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A “conspiracy theory” is a proposition that two or more people did something, such as a crime or an assassination. The term “conspiracy theory” has been cited in print since at least 1868. The Google Books Ngram Viewer shows increased usage for “conspiracy theory” from 1950.

Wikipedia: Conspiracy theory
A conspiracy theory is an explanatory proposition that accuses two or more people, a group, or an organization of having caused or covered up, through deliberate collusion, an event or phenomenon of great social, political, or economic impact. In recent decades the term has acquired a derogatory meaning, and a careful distinction must be made between the derisive use of the term and reference to actual, proven conspiracies. Numerous explanations have been proposed as to why people believe in conspiracy theories. Further, different types of conspiracy theories have been proposed. Conspiracism, a world view marked by conspiracy theories, has been elaborated, as well as its effect on society. In addition, scholars have identified some psychological origins of conspiracy theories. Finally, the socio-political origins of conspiracy theories have been analyzed, along with their deliberate use by despotic political regimes.

Usage
History
The Oxford English Dictionary records the first use of the phrase “conspiracy theory” to a 1909 article in The American Historical Review. Other sources predate this use by nearly four decades to at least 1871, where it is used in The Journal of Mental Sciencereporting on a conference of the Fifth Quarterly Meeting of the Medico-Psychological Association (now the Royal College of Psychiatrists), held on January 27, 1870:

“The theory of Dr. Sankey as to the manner in which these injuries to the chest occurred in asylums deserved our careful attention. It was at least more plausible that the conspiracy theory of Mr. Charles Beade...”

Acquired derogatory meaning
Originally a neutral term, since the mid-1960s it has acquired a somewhat derogatory meaning, implying a paranoid tendency to see the influence of some malign covert agency in events.

(Oxford English Dictionary)
conspiracy theory n. the theory that an event or phenomenon occurs as a result of a conspiracy between interested parties; spec. a belief that some covert but influential agency (typically political in motivation and oppressive in intent) is responsible for an unexplained event.
1909 Amer. Hist. Rev. 14 836 The claim that Atchison was the originator of the repeal may be termed a recrudescence of the conspiracy theory first asserted by Colonel John A. Parker of Virginia in 1880.
1952 K. R. Popper Open Society (ed. 2) II. xiv. 94, I call it the ‘conspiracy theory of society’. It is the view that an explanation of a social phenomenon consists in the discovery of the men or groups who are interested in the occurrence of this phenomenon.
1964 Listener 24 Sept. 471/3 His ‘vulgar economic realities’ are the cloak for a conspiracy-theory, and he is disappointed that I don’t subscribe to it.

16 April 1868, Boston (MA) Press and Post, pg. 1, col. 8:
The testimony of Gen. Sherman has blown the conspiracy theory of Gen. Butler to the winds; and, of course, it was in a sure anticipation of such a result that he so steadily and brazenly objected to nearly every question put by the counsel for the defence which was calculated to bring out.

Google Books
April 1870, The Journal of Mental Science, pg. 141:
It was at least more plausible that the conspiracy theory of Mr. Charles Beade, and the precautionary measure suggested by Dr. Sankey of using a padded waistcoat in recent cases of mania with general paralysis— in which mental condition nearly all of these cases under discussion were—seemed to him of practical value.

31 August 1872, Logansport (IN) , “The Energy of Despair,” pg. 2, col. 5:
Assuming that our conspiracy theory is correct, who can measure the reward, the conspirators will claim if their plot is carried out and Mr. Greeley is elected?

The original screenplay by Brian Helgeland centers on an eccentric taxi driver (Mel Gibson) who believes many world events are triggered by government conspiracies, and the U.S. Justice Department attorney (Julia Roberts) who becomes involved in his life.